Created: December 05, 2023
Modified: December 05, 2023

linear time-invariant

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

A linear time-invariant system is one where the dependence of the output $y(t)$ on the input $x(t)$ is:

linear: an input $ax(t)$ produces an output $ay(t)$ , and the sum of two inputs $x(t) + x'(t)$ produces the sum of the corresponding outputs $y(t) + y'(t)$ .
time-invariant: shifting the input in time simply shifts the output in time. $x(t - T)$ produces $y(t - T)$ .

Such a system is characterized by its impulse response $h(t)$ , which is simply the output of the system for a delta-function input. In general, the output is the convolution of the input with the impulse response:

\begin{align*} y(t) &= (x * h)(t)\\ &= \int_{-\infty}^\infty x(t-\tau) h(\tau) d\tau \\ &= \int_{-\infty}^\infty x(\tau) h(t-\tau) d\tau \end{align*}

Why is this true? We can represent any function $x(t)$ as a sum of appropriately shifted and scaled delta functions. By time-invariance, shifting produces the same response $h$ , and by linearity, scaling produces just scaled $h$ s. And again by linearity, the output from the sum of shifted-and-scaled deltas will just be the sum of these shifted-and-scaled $h$ 's.

linear time-invariant

Meta